This may seem trivial but just to confirm, as the expected value is a constant, this implies that the expectation of an expectation is just itself. It would be useful to know if this assumption is correct or if any subtleties cause this not to be true.
i.e. $ \mathbb{E[\mathbb{E[x]}]}=\mathbb{E[x]} $
This is true for the reason that you give ($\mathbb E(X)$ is constant), but in fact is a special case of a stronger and more useful result.
If $X,Y$ are two random variables then $\mathbb E(\mathbb E(X\mid Y))=\mathbb E(X)$. Here you are taking an expectation of $\mathbb E(X\mid Y)$, which is not (in general) constant but a function of $Y$.
Your original equation arises as a special case of this by taking $X,Y$ independent.
In this case $\mathbb{E[x]}$ is a constant. The expected value of the constant is the constant itself.
